Question: Find the product of the greatest common divisor and the least common multiple of $100$ and $120.$
First we find the prime factorization of each number: $$100=10\times 10=(2\times 5)\times(2\times 5)=2^2\times 5^2$$ and $$120=10\times 12=(2\times 5)\times(2\times 6)=(2\times 5)\times(2\times(2\times 3))=2^3\times 3\times 5.$$ The common factors are $2^2$ and $5,$ so $\gcd(100,120) = 2^2\times 5=20.$

The least common multiple is formed by multiplying together the highest powers of all primes occurring in the factorization of either $100$ or $120:$ $$\text{lcm}(100,120) = 2^3\times 3\times 5^2 = 8\times 3\times 25 = 24\times 25 = 600.$$ Thus, the product of the $\gcd$ and the $\text{lcm}$ is $20\times 600=\boxed{12000}.$

(Notice that this product is equal to the product of the original two numbers, $100$ and $120.$ Is that just a coincidence?)